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Local equivalence and refinements of Rasmussen's s-invariant

Nathan M. Dunfield, Robert Lipshitz, Dirk Schuetz

TL;DR

This work constructs local equivalence theories for mixed Khovanov data (LEO triples) and proves a homomorphism from the smooth knot concordance group to a local equivalence group. It develops multiple s-invariant refinements—including Bockstein- and coefficient-refined versions, plus reduced and two-reduced variants—and proves they obstruct sliceness in a broad class of knots, with strong computational verification. The framework yields a rich algebraic structure (abelian and, in a totally ordered case, a total order) and enables new concordance invariants beyond Rasmussen’s $s$, including invariants that detect triviality in LE0-classes and refine sliceness obstructions. The paper also derives structural results via conjugation on Khovanov complexes, proves relations among the refined invariants for knots, and provides substantial computational data (notably for Manolescu–Piccirillo knots) that demonstrate the practical utility of LE0 invariants in distinguishing slice from non-slice knots. Overall, the LE0 framework offers new tools for knot concordance and sliceness problems, connecting odd/even Khovanov data with Bar-Natan deformations and paving the way for further invariants derived from local equivalence concepts.

Abstract

Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.

Local equivalence and refinements of Rasmussen's s-invariant

TL;DR

This work constructs local equivalence theories for mixed Khovanov data (LEO triples) and proves a homomorphism from the smooth knot concordance group to a local equivalence group. It develops multiple s-invariant refinements—including Bockstein- and coefficient-refined versions, plus reduced and two-reduced variants—and proves they obstruct sliceness in a broad class of knots, with strong computational verification. The framework yields a rich algebraic structure (abelian and, in a totally ordered case, a total order) and enables new concordance invariants beyond Rasmussen’s , including invariants that detect triviality in LE0-classes and refine sliceness obstructions. The paper also derives structural results via conjugation on Khovanov complexes, proves relations among the refined invariants for knots, and provides substantial computational data (notably for Manolescu–Piccirillo knots) that demonstrate the practical utility of LE0 invariants in distinguishing slice from non-slice knots. Overall, the LE0 framework offers new tools for knot concordance and sliceness problems, connecting odd/even Khovanov data with Bar-Natan deformations and paving the way for further invariants derived from local equivalence concepts.

Abstract

Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group to the resulting local equivalence group of such triples. We give several versions of the -invariant that descend to , including one that completely determines whether the image of a knot in is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.
Paper Structure (21 sections, 46 theorems, 107 equations, 1 table)

This paper contains 21 sections, 46 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The even and odd Khovanov complexes and the Bar-Natan deformation
  4. Künneth theorems
  5. Even-odd local equivalence
  6. Reduced versions
  7. The s-invariant
  8. The graded integral s-invariant
  9. Invariants of LEO triples
  10. Bockstein-refined s-invariants
  11. The comprehensive refinements
  12. The reduced case
  13. Invariance and examples
  14. Structural results on reduced LEO groups
  15. The case of knots
  16. ...and 6 more sections

Key Result

Theorem 1.1

The local equivalence classes of LEO triples form an abelian group $\mathcal{C}_\mathit{LEO}$, and the Khovanov complexes induce a group homomorphism from the smooth concordance group to this local equivalence group.

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.2
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.10
  • proof
  • Lemma 2.13
  • ...and 102 more